A Tutte polynomial for toric arrangements

We introduce a multiplicity Tutte polynomial M(x,y), with applications to zonotopes and toric arrangements. We prove that M(x,y) satisfies a deletion-restriction recurrence and has positive coefficients. The characteristic polynomial and the Poincare' polynomial of a toric arrangement are shown to be specializations of the associated polynomial M(x,y), likewise the corresponding polynomials for a hyperplane arrangement are specializations of the ordinary Tutte polynomial. Furthermore, M(1,y) is the Hilbert series of the related discrete Dahmen-Micchelli space, while M(x,1) computes the volume and the number of integral points of the associated zonotope.Comment: Final version, to appear on Transactions AMS. 28 pages, 4 picture

Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity

This paper deals with the existence and the asymptotic behavior of non-negative solutions for a class of stationary Kirchhoff problems driven by a fractional integro-differential operator $\mathcal L_K$ and involving a critical nonlinearity. The main feature, as well as the main difficulty, of the analysis is the fact that the Kirchhoff function $M$ can be zero at zero, that is the problem is degenerate. The adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature

Fast Magnetic Reconnection: "Ideal" Tearing and the Hall Effect

One of the main questions in magnetic reconnection is the origin of triggering behavior with on/off properties that accounts, once it is activated, for the fast magnetic energy conversion to kinetic and thermal energies at the heart of explosive events in astrophysical and laboratory plasmas. Over the past decade progress has been made on the initiation of fast reconnection via the plasmoid instability and what has been called "ideal" tearing, which sets in once current sheets thin to a critical inverse aspect ratio $(a/L)_c$: as shown by Pucci and Velli (2014), at $(a/L)_c \sim S^{-1/3}$ the time scale for the instability to develop becomes of the order of the Alfv\'en time and independent of the Lundquist number (here defined in terms of current sheet length $L$). However, given the large values of $S$ in natural plasmas, this transition might occur for thicknesses of the inner resistive singular layer which are comparable to the ion inertial length $d_i$. When this occurs, Hall currents produce a three-dimensional quadrupole structure of magnetic field, and the dispersive waves introduced by the Hall effect accelerate the instability. Here we present a linear study showing how the "ideal" tearing mode critical aspect ratio is modified when Hall effects are taken into account, including more general scaling laws of the growth rates in terms of sheet inverse aspect ratio: the critical inverse aspect ratio is amended to $a/L \simeq (di/L)^ {0.29} (1/S)^{0.19}$, at which point the instability growth rate becomes Alfv\'enic and does not depend on either of the (small) parameters $d_i/L, 1/S$. We discuss the implications of this generalized triggering aspect ratio for recently developed phase diagrams of magnetic reconnection

Momentum Control of Humanoid Robots with Series Elastic Actuators

Humanoid robots may require a degree of compliance at the joint level for improving efficiency, shock tolerance, and safe interaction with humans. The presence of joint elasticity, however, complexifies the design of balancing and walking controllers. This paper proposes a control framework for extending momentum based controllers developed for stiff actuators to the case of series elastic actuators. The key point is to consider the motor velocities as an intermediate control input, and then apply high-gain control to stabilise the desired motor velocities achieving momentum control. Simulations carried out on a model of the robot iCub verify the soundness of the proposed approach

Automatic Gain Tuning of a Momentum Based Balancing Controller for Humanoid Robots

This paper proposes a technique for automatic gain tuning of a momentum based balancing controller for humanoid robots. The controller ensures the stabilization of the centroidal dynamics and the associated zero dynamics. Then, the closed-loop, constrained joint space dynamics is linearized and the controller's gains are chosen so as to obtain desired properties of the linearized system. Symmetry and positive definiteness constraints of gain matrices are enforced by proposing a tracker for symmetric positive definite matrices. Simulation results are carried out on the humanoid robot iCub.Comment: Accepted at IEEE-RAS International Conference on Humanoid Robots (HUMANOIDS). 201

Multiband superconductors close to a 3D-2D electronic topological transition

Within the two-band model of superconductivity, we study the dependence of the critical temperature Tc and of the isotope exponent alpha in the proximity to an electronic topological transition (ETT). The ETT is associated with a 3D-2D crossover of the Fermi surface of one of the two bands: the sigma subband of the diborides. Our results agree with the observed dependence of Tc on Mg content in A_{1-x}Mg_xB_2 (A=Al or Sc), where an enhancement of Tc can be interpreted as due to the proximity to a "shape resonance". Moreover we have calculated a possible variation of the isotope effect on the superconducting critical temperature by tuning the chemical potential.Comment: J. Supercond., to appea